Intersection multiplicity of Serre on regular schemes

Journal of Algebra 319 (2008) 1530–1554 www.elsevier.com/locate/jalgebra

Intersection multiplicity of Serre on regular schemes S.P. Dutta 1 Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA Received 18 January 2007 Available online 26 November 2007 Communicated by Luchezar L. Avramov

Abstract The study of the intersection multiplicity function χ OX (F , G) over a regular scheme X for a pair of coherent OX -modules F and G is the main focus of this paper. We mostly concentrate on projective schemes, vector bundles over projective schemes, regular local rings and their blow-ups at the closed point. We prove that (a) vanishing holds in all the above cases, (b) positivity holds over Proj of a graded ring finitely generated over its 0th component which is artinian local, when one of F and G has a finite resolution by direct sum of copies of O(t) for various t, and (c) non-negativity holds over PnR , R regular local, and over arbitrary smooth projective varieties if their tangent bundles are generated by global sections. We establish a local– global relation for χ for a pair of modules over a regular local ring via χ of their corresponding tangent cones and χ of their corresponding blow-ups. A new proof of vanishing and a special case of positivity for Serre’s Conjecture are also derived via this approach. We also demonstrate that the study of non-negativity is much more complicated over blow-ups, particularly in the mixed characteristics. © 2007 Elsevier Inc. All rights reserved. Keywords: Intersection multiplicity; Hilbert function; Vector bundle; Sheaf cohomology; Dimension

Introduction Let (R, m, K) be a regular local ring of dimension n, i.e., m is the maximal ideal of a regular local ring R and K = R/m. Let M and N be two finitely generated R-modules such  that (M ⊗R N) < ∞. Serre introduced the notion of χ(M, N) as (−1)i (TorR i (M, N )) (“” stands for length) and conjectured that (i) χ(M, N)  0 and (ii) χ(M, N) = 0 if and only E-mail address: [email protected]. 1 Supported by an NSA grant MDA H98230-04-1-0100.

0021-8693/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2007.10.016

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if dim M + dim N < dim R (see Brief History for more details). In the mid-nineties Gabber [G] came up with a brilliant idea to prove part (i) of this conjecture. The key steps in his proof are the following: (a) use de Jong’s Theorem on regular alteration to reduce χ(R/P , R/q) to     χ OX (OY  , OZ  ) where X  = PN R , Y , Z are closed subvarieties of X such that Z is regular and O



support of Tori X (OY  , OZ  ) ⊂ PN K for i  0, (b) extend a spectral sequence argument of Serre to reduce the problem to χ of intersection of a closed subscheme of the normal bundle corresponding to the regular imbedding Z  → X  with the 0-section of this bundle, (c) use ramification of R and module of differentials to show that the closed fiber E over s = [m] of this bundle is generated by global sections (superb ingenious technique!) and (d) establish non-negativity of intersection multiplicity on vector bundles over projective schemes generated by global sections. We call this whole procedure “Gabber’s technique” and use several parts of this technique in Sections 2 and 3. One of the main aims of this paper is to see how far we can push and extend this technique to achieve substantial results on intersection multiplicity, as introduced by Serre, not only for regular local rings but also for regular schemes as well. For a good exposition of Gabber’s proof we refer the reader to [Ho2]. We define intersection multiplicity on a regular variety X over an excellent discrete valuation ring V or over a field k. Let F and G be two coherent OX -modules. Suppose that X (H i (X, TorO j (F, G))) < ∞ for i, j  0. We define χ OX (F, G) =

    X (−1)i+j  H i X, TorO j (F, G)

(Grothendieck’s definition of hypercohomology, EGA III). This definition was used by Fulton and MacPherson in [Fu-M] where they used Grothendieck’s Riemann–Rock Theorem and diagonalization technique to prove vanishing over smooth projective varieties over algebraically closed fields (see part (a) of Theorem 3). Theorem 2, Proposition 2.2, Theorems 3–5 are the main results of this paper. Theorem 1 and one of its corollaries are used for proving Theorem 2. Several results from [Fu] are utilized in the proofs of Proposition 2.2 and Theorem 5, respectively. Let me describe the main results of this paper briefly. In Section 1, first we state an extension of a theorem of Peskine and Szpiro [P-S2] and prove the following.  Theorem 1. Let B = i0 Bi , be a noetherian graded ring such that Bo is artinian local, and let M and N be two finitely generated graded B-modules. Suppose that M has a finite resolution oflength s by graded free modules consisting of homomorphisms of degree 0. Write Pχ(M,N) = si=0 (−1)i PTorB (M,N ) . i Then we have the following: (a) Pχ(M,N)  0, the sign of equality holds if and only if dim M + dim N < dim B, and (b) if dim M + dim N  dim B, then Pχ(M,N) is an essential polynomial of degree r = dim M + dim N − dim B with leading co-efficient e(M)e(N )/r!e(B), where e(T ) denotes the Hilbert multiplicity of any finitely generated graded module T . As a corollary we derive:

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 Corollary 1. (To Theorem 1.) Let X = Proj B, B = n0 Bn be a noetherian graded ring such that Bo is artinian local and B is generated by finitely many elements of degree 1 as a Bo ˜ G = N˜ ; M, N being two algebra. Let F, G be two coherent OX -modules such that F = M, finitely generated graded B-modules. Suppose that M has a finite free graded resolution over B. Then χ OX (F(t), G) is a polynomial function for all t ∈ Z such that (i) χ OX (F(t), G)  0 for t  0; the sign equality holds for all t if and only if dim Supp F + dim Supp G < dim X, and (ii) if dim Supp F + dim Supp G = dim X, then χ OX (F(t), G) = e(M)e(N )/e(B) for all t. Here e(M) (etc.) denotes Hilbert multiplicity of M. This immediately leads to the following: Corollary 2. Let k be a field and X = Pnk . Let F, G be two OX -modules. Then χ OX (F(t), G) is a polynomial function for all t ∈ Z such that (i) χ OX (F(t), G)  0 for t  0, sign of equality holds for all t if and only if dim Supp F + dim Supp G < dim X, and (ii) if dim Supp F + dim Supp G = dim X, then χ OX (F(t), G) = e(M)e(N ) for all t ∈ Z; M, N as in Corollary 1. The above corollary can also be deduced via intersection product, Riemann–Roch Theorem and diagonalization technique (see [Fu, Chapters 8, 18]). We use Theorem 1 and Corollary 2 in our proof of the main theorem in Section 1. To describe our main theorem in Section 1 we need the following set-up: R as above and, let P , q be two prime ideals of R such that (R/(P + q)) < ∞. Write X = Spec R, Y = Spec(R/P ), Z = Spec(R/q). Let π : X˜ → X be the blow-up of X at {m}, E the exceptional divisor and η : E → {m} the induced map. Since R is regular local of dimension n, E = Pn−1 K . Let Y˜ , Z˜ denote the blow-ups of Y and Z respectively at {m}. Since (R/(P + q)) < ∞, Y˜ ∩ Z˜ ⊂ E. ˜ TorOX˜ (O ˜ , O ˜ ))) is finite for i, j  0. For any finitely generated R-module M, Hence (H i (X, j Y Z em (M) denotes the Hilbert–Samuel multiplicity of M, G(M) denotes the graded module corresponding to the m-adic filtration on M, and for any pair M, N of such modules, Pχ(G(M),G(N )) denotes the alternating sum of Hilbert polynomials PTori (G(M),G(N )) (t), for t  0, on the graded ring G(R). Now we can state our next theorem. Theorem 2. With the above set-up we have the following: (i) if dim R/P + dim R/q = dim R, then χ(R/P , R/q) = χ OX˜ (OY˜ , OZ˜ ) + Pχ(G(R/P ),G(R/q)) and (ii) if dim R/P + dim R/q < dim R, then χ(R/P , R/q) = χ OX˜ (OY˜ , OZ˜ ). As a corollary, first we deduce that in case (i), χ(R/P , R/q) = em (R/P )em (R/q) + χ OX˜ (OY˜ , OZ˜ ) and then we deduce, via Serre’s Theorem (see Brief History), that when R

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is equicharacteristic, χ OX˜ (OY˜ , OZ˜ )  0. A completely different geometric proof of this nonnegativity is sketched in Remark 4 at the end of Section 3.2. This corollary has been mentioned in Fulton’s book [Fu, Example 20.4.3]; a proof using local Chern characters and Riemann–Roch Theorem was presented in [D5]. Here, our theorem connects local and global χ directly via the tangent cone and the blow-up X˜ of the local ring without using any heavy machinery and it formulates what happens in the vanishing part which is completely new. Thus this theorem provides a new approach to Serre’s conjecture which we will exploit in Section 3. In Section 2, we study χ OX (F, G) where X represents a projective scheme. Recall that in order to prove non-negativity Gabber required the vector bundle E to be generated by global sections. In this section we would like to understand the situation where E is not necessarily so. We prove that vanishing holds even when E is not generated by global sections. This plays an important role in proving vanishing over blow-ups. We have the following proposition. Proposition 2.2. Let W be a projective scheme over a field and L be a locally free OW -module of rank d. Let E = Spec(SymOW (L)) and let V be a closed subscheme E. Let β : W → E denote the O-section; we identify W with β(W ) when there is no scope of ambiguity. We have the following:  (i) χ OE (OV , OW ) = V¯ cd (ξV¯ )td(ξV¯ )−1 τ (V¯ ), where ξV¯ is the restriction of the universal quotient bundle ξ on P(E ⊕ 1) to the projective closure V¯ of V ; (ii) if dim V < d, then χ OE (OV , OW ) = 0; (iii) if V ∩ W = φ, then χ OE (OV , OW ) = 0; and   (iv) if dim V = d, then χ OE (OV , OW ) = V¯ cd (ξV¯ )[V¯ ] = β ∗ ([V ]). Part (iv) of the above proposition along with Theorem 12.1(a) in [Fu] provide a new proof of Gabber’s proposition on non-negativity over vector bundles generated by global sections. We use the above proposition and Gabber’s technique to prove our next theorem. Part (a) of this theorem can also be deduced by using intersection cycles, diagonalization and Riemann– Roch Theorem [Fu-M]. Theorem 3. Let X be a smooth projective variety over a perfect field k. Let Y and Z be two closed subvarieties of X such that dim Y + dim Z  dim X. We have the following: (a) If dim Y + dim Z < dim X, then χ OX (OY , OZ ) = 0. (b) If TX is generated by global sections and dim Y + dim Z = dim X, then χ OX (OY , OZ )  0. Here TX is the tangent bundle on X. In our follow-up remark we point out the existence of counter-examples to part (b) of the above theorem when TX is not generated by global sections. This answers a question raised by M. Hochster. In our final theorem of this section we prove that both vanishing and non-negativity is valid on PnR . Theorem 4. Let Y and Z be two closed subschemes of P = PdR such that Y ∩ Z ⊂ Pds = PdK . Then (i) if dim Y + dim Z < dim P, χ OP (OY , OZ ) = 0 and (ii) if dim Y + dim Z = dim P, χ OP (OY , OZ )  0.

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We next investigate the effect on intersection multiplicity created by a pull-back map on vector-bundles. We prove the following: Proposition. Let W be a smooth complete scheme over a field K and let ϕ˜ : E  → E be a OW -map between two locally free OW -modules of ranks r  and r respectively. Write E  = Spec(SymOW (E  )) and E = Spec(SymOW (E)) and ϕ : E → E  denote the map of vector bundles corresponding to ϕ. ˜ Let V be a closed subscheme of E  such that the dimension of support OE  of Tori (OV , OE ) < r for i > 0. Let F be a coherent OW -module. Then χ OE  (OV , F) = χ OE (Oϕ −1 (V ) , F). As a corollary we derive Corollary. Let X be a smooth projective variety over a field K such that TX is generated by global sections. Let W be a smooth closed subvariety of codimension r. Let I denote the ideal of definition of W in X and let E be the corresponding normal cone. Let E  be a vector bundle on W and let V be a closed subvariety of E  . Suppose that there exists a map ϕ : E → E  of O  bundles such that the dimension of support of Tori E (OV , OE ) < r for i > 0. Then, for any coherent sheaf F on W , χ OE  (OV , F) = χ OE (Oϕ −1 (V ) , F). Moreover, if dim ϕ −1 (V )  r, then χ OE  (OV , OW )  0. ˜ Here we use several In Section 3 we take up the issue of intersection multiplicity over X. results from Chapters 3, 6, 12, and 18 in [Fu]. To describe our next theorem we continue with the set-up for Theorem 2. Assume that R is of essentially finite type over a field or an excellent discrete valuation ring. Let X˜ denote blow-up of X at s = [m] and let Y˜ , Z˜ be two closed subva ˜ rieties of X˜ such that Y˜ ∩ Z˜ ⊂ E = Pn−1 K (= the fiber over s) in X. Let Z be a regular alteration N   ˜ Then Z has a closed immersion in P = X for some N > 0. Let π : X  → X˜ be the proof Z. X˜   Suppose jection map and write Y  = π −1 (Y˜ ). Let I denote the sheaf of ideals defining  Z t in X . t+1   2 r = codim of Z in X . Let E = Spec(SymOZ (I /I )) and V = Spec( t0 I OY  /I OY  ). Let E1 , . . . , Ed , V1 , . . . , Ve , and W1 , . . . , Wd denote the components of Es (fiber over s), V and Zs respectively. Write Eαβ to denote the restriction of E to Wα ∩ Wβ . Now we are ready to state our final theorem. ˜ Y˜ , Z˜ etc. as above. Suppose that dim Y˜ + dim Z˜  dim X. ˜ We have the Theorem 5. Let X, following: (i) if dim Y˜ + dim Z˜ < dim X˜ or Y˜ ∩ Z˜ = φ, χ OX˜ (OY˜ , OZ˜ ) = 0. (ii) Let Y˜ and Z˜ be blow-ups of Y and Z at their respective closed points. If dim G(R/P ) ⊗ G(R/q)  1, then χ OX˜ (OY˜ , OZ˜ )  0. (iii) There exists sub-bundles Fαβ of rank (r − 1) of Eαβ , 1  α, β  d, such that if any Vi is contained in any Fαβ , then χ OE (OVi , OZs )  0. As a corollary we derive a completely new proof of Serre-vanishing and a proof of a special case of positivity, i.e., χ(M, N)  em (M)em (N ), if dim G(M) ⊗ G(N )  1. Part (iii) of this theorem demonstrates that even if χ over blow-up is non-negative (Corollary 2, Theorem 2), the possibility of having negative intersection multiplicity for certain component of a subcone of the normal bundle originating from blow-up does exist.

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In our remarks following the proof we point out that the proof in part (iii) shows that Es , constructed as above, is not generated by global sections. Thus, Gabber’s technique cannot be extended from the local case to blow-ups even when none of the subvarieties under consideration is contained in the exceptional divisor. Finally, in Remarks 3, we sketch a geometric proof, totally different from the proof outlined in the corollary of Theorem 2, of the fact that when R is equicharacteristic χ OX˜ (OY˜ , OZ˜ )  0 whenever dim Y + dim Z = dim X. This proof is inspired by ideas in Chapter 12 of [Fu]. Brief history Serre proved his conjecture for equicharacteristic and unramified regular local rings [S1, Theorem 1, Chapter V]. He also proved that over any regular local ring R, if M and N are two finitely generated modules such that (M ⊗R N ) < ∞, then dim M + dim N  dim R; moreover if dim M + dim N = dim R, then χ(M, N)  em (M)em (N ). The conjecture can be divided into three parts: (a) vanishing: χ(M, N) = 0 when dim M + dim N < dim R, (b) non-negativity: χ(M, N)  0, and (c) positivity: χ(M, N) > 0 when dim M + dim N = dim R. P. Roberts [R1], H. Gillet and C. Soulé [G-S] proved the vanishing part independently (in mid-eighties). Their proofs extend to the local complete intersections when both M and N have finite projective dimension. The hope of generalizing the validity of this conjecture to non-regular rings when only one of the modules has finite projective dimension was dashed by a counterexample due to Hochster, McLaughlin and this author [D-H-M] in the early eighties. This example also led to counter-examples to several other multiplicity related conjectures. In the mid-nineties Gabber [B] proved the non-negativity part of the conjecture. The positivity part of Serre’s conjecture has been open for almost fifty years. The fact that positivity or non-negativity implies vanishing was proved in [D1] in the early eighties. For notations and terminology in Commutative Algebra and Algebraic Geometry we refer the reader to [Ma] and [H] respectively. For Chern class, local Chern character, intersection product and related matters we refer the reader to [Fu]. By dim R (or dim X) we mean the dimension of R (or dimension of X). While studying intersection multiplicity on a regular local ring (R, m, K) we can assume without any loss of generality that K is algebraically closed and the modules M, N can be replaced by R/P , and R/q respectively where P and q are prime ideals in R. For any coherent sheaf F on a scheme X, P(F) denotes the projective scheme Proj(SymOX (F)) and F ∨ stands for HomOX (F, OX ). 1. In this section, first we will prove an extension of a theorem of Peskine and Szpiro [P-S2, Theorem 2] and derive some corollaries in the geometric set-up.  1.1. Let B = n0 Bn be a noetherian graded ring such that B0 is artinian local. Let PM denote the Hilbert polynomial for a finitely generated  graded B-module M and let e(M) denote the Hilbert multiplicity of M. Recall that PM (n) = i
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Lemma. Let T• be a finite complex of finitely generated graded free B-modules with homomorphisms of degree 0 T• : 0 →

ts 

B[−dsj ] → · · · →

j =1

t1 

B[−d1j ] →

j =1

t0 

B[−d0j ] → 0.

j =0

Then we have the following   (−1)k ρk PB(k) (−1)i PHi (T• ) = k! i0

where ρk =



i i0 (−1)

 ti

k j =1 dij

k0

(k)

and PB is the kth derivative of PB .

For a proof we refer the reader to [P-S2]. 1.2.

Now we state and prove an extension of Theorem 2 in [P-S2].

Theorem 1. Let B be as above and let M and N be two finitely generated graded B-modules. Suppose that M has a finite resolution of length s by graded free modules consisting of homomorphisms of degree 0. Write Pχ(M,N) = si=0 (−1)i PTorB (M,N ) . i Then we have the following: (a) Pχ(M,N)  0, the sign of equality holds if and only if dim M + dim N < dim B, and (b) if dim M + dim N  dim B, then Pχ(M,N) is an essential polynomial of degree r = dim M + dim N − dim B with leading co-efficient e(M)e(N )/e(B)r!, where e(T ) denotes the Hilbert multiplicity of any finitely generated graded module T . Proof. The arguments are essentially the same as in the proof of Theorem 2 in [P-S2]. Hence we leave this as an exercise to the reader. 2 This theorem leads to the following corollaries. Corollary 1. Let X = Proj B, with B as in the theorem. Suppose that B is generated by B1 as a B0 -algebra. Let F and G be two coherent OX˜ -module such that F = M˜ and G = N˜ where M and N are two finitely generated graded B-modules. Suppose that M has a finite graded free resolution of length s (as in Theorem 1).  X OX (F(t), G) is a Let χ OX (F(t), G) = (−1)i+j length H i (X, TorO j (F(t), G)). Then χ O polynomial function, i.e. ∃ a polynomial P ∈ Q[T ] s.t. χ X (F(t), G) = P (t), ∀t ∈ Z (Hilbert Polynomial, EGA III, 2.5; §80, [S2]) such that (i) for t  0, χ OX (F(t), G)  0, the sign of equality holds for all t if and only if dim Supp F + dim Supp G < dim X, (ii) if dim Supp F + dim Supp G = dim X, χ OX (F(t), G) = e(M)e(N )/e(B) for every t.

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OX X Proof. It is easy to check that TorO i (F(t), G) = Tori (F, G)(t). Then the assertion that χ OX (F(t), G) is a polynomial function follows from Theorem 2.5.3, EGA III. Taking G = N˜ , ¯ it is easy to see that where N = B/P (= B),

  χ OX F(t), G = Pχ(M,N) (t) = Pχ(M,N) (t + 1) − Pχ(M,N) (t). 2

Hence the result follows from the above theorem.

Corollary 2. Let K be a field and let X = PnK . Let F , G be two coherent OX -modules. We have the following: (i) χ OX (F(t), G)  0 for t  0, sign of equality holds for all t if and only if dim Supp F + dim Supp G < dim X and (ii) if dim Supp F + dim Supp G = dim X, then χ OX (F(t), G) = e(M)e(N ) for all t ∈ Z (M, N as in Corollary 1). Proof follows immediately from Corollary 1 and the fact that e(B) = 1, since B is a polynomial ring over K. The above lemma, Theorem 1 and Corollary 2 are used in the proof of our next theorem. 1.3. Let (R, m) be a local ring and let M, N be two filtered R-modules with m-good filtrations (i.e., if (Mn ) is an m-adic filtration of M, then ∃n0 > 0 s.t. mh Mn = Mn+h , ∀n  n0 and any h > 0; similarly for N ). We say f : M → N a filtered map if f is R-linear and f (Mn ) ⊂ Nn and f is strict if f(Mn ) = f (M) ∩ Nn . Let G(M) denote the corresponding graded module for M, i.e. G(M) = n0 Mn /Mn+1 . It is easy to check that if f

g

M− →N − →P is exact and f and g are strict, then G(f )

G(g)

G(M) −−−→ G(N) −−−→ G(P ) is also exact. Let R be given m-adic filtration by {mn }, i.e., R = R0 ⊃ R1 ⊃ · · · ⊃ Rn ⊃ · · · , where Rn = mn . Denote by R(i) the filtered R-module defined by R(i)n = Rn−i , ∀n (here Rj = R if j  0). It is easy to see that R(i) is a filtered free R-module  of rank 1. Moreover, if f : F → M is a strict surjective map of filtered modules and F = si=1 R(−ti ), then Fr → Mr is onto for r  max{ti | 1  i  s}. Now assume R is regular. Then given any finitely generated filtered module M, one can find a filtered free resolution F• of M F• : 0 → Fs → Fs−1 → · · · → F1 → F0 → 0 where each Fi is a finite direct sum of filtered free modules of the type R(i) as defined above and the maps are strict. One can easily check that this resolution is nothing but a lift of a minimal free graded resolution of G(M) over G(R). Since R is regular local, G(R) is a polynomial ring in n variables when n = dim R. Note that such a filtered free resolution need not be minimal.

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Recall that if M has filtration (M n ), N has filtration (Nn ), then we endow M ⊗ N with filtration imposed by (M ⊗R N )n = i+j =n Im(Mi ⊗ Nj ). Now we are ready to state and prove our next proposition. Proposition. Let R be regular local and let M and N be two finitely generated R-modules such (M ⊗R N ) < ∞. We have the following: (a) if dim M + dim N = dim R, then χ(M, N) = c + Pχ(G(M),G(N )) , where the integer c is associated to Serre’s spectral sequence arising from F• ⊗ N , F• being a filtered free resolution of M (here we assign filtrations Mn = mn M on M and Nn = mn N on N ) as described above, and (b) if dim M + dim N < dim R, then χ(M, N) = c. Proof. Let us endow M(N ) with filtration {Mn = mn M}({Nn = mn N }). Write C• = F• ⊗ N , where F• is a filtered free resolution of M as described above. Then C• is a finite filtered complex. Let us denote by Er , 0  r  ∞, the spectral sequence associated to C• (corresponding to the product filtration on each component Fi ⊗ N of C• ). It is easy to see that G(R) i is the associated graded module of TorR (M, N ) (inherited E1i = Tori (G(M), G(N)) and E∞ i from the m-good filtration of the above complex). Serre has shown that this spectral sequence {Er } is convergent [S1, Théorème, Chapter II, p. 22; Chapter IV, §3]. Since dim G(M) + dim G(N) = dim M + dim N  dim R = dim G(R), by Theorem 1, we obtain em (M)em (N)/em (R) = em (M)em (N ), if dim M + dim N = dim R, Pχ(G(M),G(N )) = 0, otherwise (R being regular local, em (R) = 1). Since Pχ(G(M),G(N )) is constant, there exists a positive integer n0 such that for t  n0 , Pχ(G(M),G(N )) (t) is constant. Let p be an integer greater than n0 . Denote by (C• )i the ith filtration of C• . Write C˜ • = C• /(C• )p+1 . Then C˜ • is a filtered complex of modules of finite i i length such that E1,j of C˜ • is the same as E1,j of C• for j  p. Since (Hi (C˜ • )) < ∞ and (E i (C˜ • )) < ∞ for 1  i  s, we have 1

s s s        (−1)i  Hi (C˜ • ) = (−1)i  E1i = (−1)i PTorG(R) (G(M),G(N )) i=0

i=0

i=0



= Pχ(G(M),G(N )) =

i

em (M)em (N ),

if dim M + dim N = dim R,

0

if dim M + dim N < dim R

by Theorem 1. Now, let us consider the exact sequence O → (C• )p+1 → C• → C˜ • → 0. From the long exact sequence of homologies, we obtain

S.P. Dutta / Journal of Algebra 319 (2008) 1530–1554

χ(M, N) =

1539

           (−1)i  Hi (C˜ • ) (−1)i  Hi (C• ) = (−1)i  Hi (C• )p+1 +

= c + Pχ(G(M),C(N )) where c =

 (−1)i (Hi ((C• )p+1 )). Hence the theorem is proved.

Corollary. With hypothesis as in case (a), we have χ(M, N) = c + em (M)em (N ). 1.4. Let (R, m, K) denote a regular local ring of dimension n, essentially of finite type over a field or a discrete valuation ring V . Let P and q be two prime ideals such that (R/(P +q)) < ∞. Write X = Spec R, Y = Spec(R/P ), Z = Spec(R/q). Let π : X˜ → X be the blow-up of X at {m}, E the exceptional divisor and η : E → {m} the induced map. Since R is a regular local ˜ ˜ ring of dim n, E = Pn−1 K , K = R/m. Let Y , Z denote the blow-ups of Y and Z respectively ˜ TorOX˜ (O ˜ , O ˜ ))) is finite. Let at {m}. Since (R/(P + q)) < ∞, Y˜ ∩ Z˜ ⊂ E. Hence (H i (X, j

Y

Z

Z¯ = E ∩ Z˜ and Y¯ = E ∩ Y˜ .  O Let χ OX˜ (OY˜ , OZ˜ ) = (−1)i+j (H i (Torj X˜ (OY˜ , OZ˜ ))). Now we are ready to state our next theorem. Theorem 2. Let R, P , q, etc. as above. We have the following: (i) if dim R/P + dim R/q = dim R, then χ(R/P , R/q) = Pχ(G(M),G(N )) + χ OX˜ (OY˜ , OZ˜ ) and (ii) if dim R/P + dim R/q < dim R, then χ(R/P , R/q) = χ OX˜ (OY˜ , OZ˜ ). Proof. We have, by the previous theorem, χ(R/P , R/q) = c + em (R/P )em (R/q).

We need to show that c = χ OX˜ (OY˜ , OX˜ ). Then the previous theorem will establish the required result. O Since (R/(P + q)) < ∞, for t  0, mt Tori X˜ (OY˜ , OZ˜ ) = 0 for i  0. Hence these Tor modules can be viewed as coherent sheaves on X˜ ×Spec R Spec(R/mt ). Consider the exact sequence o → OX˜ (1) → OX˜ → OP → 0,

P = Pn−1 K .

For any t ∈ Z, we obtain 0 → OZ˜ (t) → OZ˜ (t − 1) → OZ¯ (t − 1) → 0. By Corollary 2, Theorem 1, we obtain χ OX˜ (OY˜ , OZ˜ (t)) = χ OX˜ (OY˜ , OZ˜ (t − 1)). Hence χ OX˜ (OY˜ , OZ˜ (t)) is a constant for t ∈ Z.

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Let F• be a free resolution of R/P over R. Then π ∗ (F• ) is a finite free complex of   r OX˜ -modules such that Hi (π ∗ (F• )) = TorR r0 m . Note that i (R/P , S), i  0; here S =  H0 (π ∗ (F• ))  S/P S.  Let T denote the finitely generated graded S-module r0 Tor1 (R/P , R/mr ). By Artin– Rees Lemma, there exists h > 0 such that mh T = 0 and mh TorR i (R/P , S) = 0 for i > 0. Hence T and TorR (R/P , S), for i > 0, has a filtration whose successive quotients are finitely generi R r ated Gm (R) module. Note that, for i  0, (Tori (R/P , R/m )) is an essentially polynomial function—denote it by fi (r). Hence, by the above observation, lemma, Theorem 1 and Corollary 2, we have, for i > 0. χ



TorR i

     r R/P , R/m , OZ˜ (t) = em (R/q)Δs fi (t) + lower degree terms in t,

r0

s = dim R − dim R/q = dim R/P .

(1)

Let G• denote the complex π ∗ (F• ) ⊗ OZ˜ (t). Tensoring a finite resolution of OZ˜ (t) by lo2 cally free OX˜ -modules with π ∗ (F• ), we obtain a spectral sequence whose Ep,q terms are OX˜ Torp (Torq (R/P , S), O ˜ (t)); this spectral sequence converges to Hp+q (G• ). Hence Z

       ˜ Hj (G• ) = (−1)i+j  H i X, (−1)i χ OX˜ Tori (R/P , S), OZ˜ (t) .

(2)

i0

Since π is proper, the left-hand side of the above equation equals     χ R/P , mt + q /q = χ(R/P , R/q).

(3)

Now consider the exact sequence  0 → T˜ → S/P S → OY˜ → 0 we have        S, OZ˜ (t) − χ OX˜ T˜ , OZ˜ (t) = χ OX˜ OY˜ , OZ˜ (t) . χ OX˜ S/P

(4)

Since   χ R/P , R/mr = 0,

     χ1 R/P , R/mr =  R/ P + mr .

(5)

Hence      χ OX˜ T˜ , OZ˜ (t) + (−1)i χ OX˜ Tori (R/P , S), OZ˜ (t) i1

= em (R/P )em (R/q)

  by (1) and (5) .

(6)

S.P. Dutta / Journal of Algebra 319 (2008) 1530–1554

Thus, from (2)–(4) and (6), it follows that   χ OX˜ OY˜ , OZ˜ (t) = χ R (R/P , R/q) − em (R/P )em (R/q) = c.

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2

Corollary 1. With hypothesis as in case (i), we have χ(R/P , R/q) = em (R/P )em (R/q) + χ OX˜ (OY˜ , OZ˜ ). Corollary 2. R equicharacteristic regular local ring. Then χ OX˜ (OY˜ , OZ˜ )  0. The proof follows from Serre’s Theorem (mentioned in the introduction) and the above result. 2. 2.1. First we would like to state a proposition from Gabber’s work which plays an important role in his proof of non-negativity. Proposition. Let W be a complete scheme over a field. Let L be a local free OW -module of rank r such that L∨ = HomOW (L, OW ) is generated by global sections. Write E = Spec(SymOW (L)). Let V be a closed subscheme of E such that dim V  r. Let β : W → E be the zero-section and we identify β(W ) with W . Then χ OE (OW , OV )  0. For a proof, we refer the reader to Proposition 6.1.6 in [B] or Step 5 in [Ho2]. We will use this proposition several times in our work. Our next proposition shows that for vanishing we do not need E to be generated by global sections. 2.2. Proposition. Let W be a projective scheme over a field and let L be a locally free OW module of rank d. Let E = Spec(SymOW (L)) and let V be a closed subscheme of E. Let β : W → E be the 0-section; we identify W with β(W ) when there is no scope of ambiguity. We have the following:  (i) χ OE (OV , OW ) = V¯ cd (ξV¯ )t d(ξV¯ )−1 ∩ τ (V¯ ), where ξV¯ is the restriction of the universal quotient bundle ξ on P(E ⊕ 1) to the projective closure V¯ of V ; (ii) if dim V < d, then χ OE (OV , OW ) = 0; (iii) if V ∩ W = φ, then χ OE (OV , OW ) =0; and  (iv) if dim V = d, then χ OE (OV , OW ) = V¯ cd (ξV¯ ) ∩ [V¯ ] = β ∗ ([V ]). Proof. Let P = P(L); Q = P(L ⊕ OW )—the projective closure of E. Then E ∩ P = φ. Let V¯ = closure of V in Q. We have the following commutative diagram: i

P

j

Q

E β

q p

W

(1) π

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and the following exact sequence    0 → OQ (−1) → q ∗ L∨ ⊕ OW → ξ → 0

(2)

where ξ is a locally free OQ module of rank d. The composite q ∗ (1) → q ∗ ((L∨ ) ⊕ OW ) → ξ gives a regular section t of ξ which vanishes exactly on the zero-section β(W ). Write f = j ◦ β. We have a free OQ resolution of f∗ OW by exterior powers of ξ (Koszul resolution) as follows: 0→

d

ξ∨ →

d−1

∨

t → OQ → 0. ξ∨ → ··· → ξ∨ −

(3)

This is the Koszul complex determined by the regular section t. Let us denote the pull back of ξ over V¯ by ξV¯ . By tensoring (3) with OV¯ , we obtain the following sequence 0→

d

ξV∨¯ → OQ

The homologies of (4) are Tori

d−1

ξV∨¯ → · · · → ξV∨¯ → OV¯ → 0.

(4)

(OV¯ , OW ). By considering a resolution of OV¯ by loO

E cally free OQ -modules it is easy to see that Tori Q (OV¯ , OW )  TorO i (OV , OW ) and hence O O χ Q (OV¯ , OW ) = χ E (OV , OW ). ξV¯ is locally free sheaf of rank d over V¯ and V¯ is a complete scheme of dimension  d. By Example (18.3.7) in [Fu] we have the following:

   O (−1)i+j dimK H i Q, Torj Q (OV¯ , OW ) =

  

j j  (−1)j χ V¯ , (−1)i+j dimK H i V¯ , ξV∨¯ = ξV∨¯

=



 j ch ξV∨¯ ∩ τ (V¯ ) (−1)j

V¯

=

cd (ξV¯ )td(ξV¯ )−1 ∩ τ (V¯ )

 [Fu, Riemann–Roch Theorem, 18.3])

  [Fu, Example 3.2.5] .

(5)

V¯

 If dim V¯ < d = rank of ξ , then V¯ cd (ξV¯ ) = 0. Thus the vanishing holds. Moreover, if  V¯ ∩ W = φ, then ξV¯ has a no-where vanishing section on V¯ and hence V¯ cd (ξV¯ ) = 0. Thus χ OE (OW , OV ) = 0. If dim V = d, then τ (V¯ ) and V¯ agree in dimension d and the first equality of (iv) is immediate; the second one follows from Proposition (3.3) in [Fu] (for statement of this proposition see Proposition 1 in Section 3). 2 Remark 1. Part (iv) of this proposition along with Theorem 12.1(a) in [Fu] offer another proof of Gabber’s result on non-negativity of intersection multiplicity of χ OE (OV , OW ) when dim V  rank E and E is generated by global sections.

S.P. Dutta / Journal of Algebra 319 (2008) 1530–1554

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Remark 2. The same proof works if instead of assuming W to be complete, we assume that V is a complete closed subscheme of E. It also works in the mixed characteristics provided V is proper over a field k. The above proposition is used in the proofs of part (a) of the following theorem and part (i) of Theorem 5. Theorem 3. Let X be a smooth projective variety over a perfect field K. Let Y and Z be two closed subvarieties of X such that dim Y + dim Z  dim X. Then we have the following (a) if dim Y + dim Z < dim X, then χ OX (OY , OZ ) = 0, (b) if TX is generated by global sections and dim Y + dim Z = dim X, then χ OX (OY , OZ )  0. Here TX is the tangent bundle on X. Proof. Let Z  → Z be a regular alternation [J, Theorem 4.1]. You may assume that Z  → Proj(OZ [T1 , . . . , TN +1 ]). Write X  = Proj(OX [T1 , . . . , TN +1 ]). Then we have the following commutative diagram Z

X η

Z

X.

Since X  is smooth (X being so) and Z  is smooth (K is perfect), Z  → X  is a regular imbedding of codimension d say, i.e., Z  = V (I) where I ⊂ OX is a locally complete intersection sheaf of ideals of height d (I is locally generated by a regular sequence of length d). Let Y  = η−1 (Y ). Since X   X is flat of relative dimension N , dim Y  = dim Y + N . Since dim Z  = dim Z, dim Y  + dim Z   dim X  and dim Y  + dim Z  < dim X  if and only if dim Y + dim Z < dim X. Since η is proper, by pulling back a free OX -resolution of OY over X via η, we obtain, by projection formula, χ OX (OY  , OZ  ) = χ OX (OY , η∗ OZ  ) +

   (−1)i χ OX OY , R i η∗ OZ  .

(1)

i1

Since dim support of R i η∗ OZ  < dim Z and since η∗ OZ has a filtration by a finite number of copies of OZ and OZi where dim OZi < dim OZ , by inducting on dim Z, it is enough to prove our theorem for Y  , Z  in X  . Now we use the sheafification of Serre’s spectral sequence on regular sequences as demonstrated by Gabber [B]. Let E = Spec(OX /I ⊕ I/I 2 ⊕ · · ·). Then E is a vector bundle of rank d over Z  . Write Z  = W and let β : W → E be the zero-section. Let V = Spec(OY  /IOY  ⊕ IOY  /I 2 OY  ⊕ · · ·). Then dim E = dim Z  + d = dim X  , dim V = dim Y  . By a spectral sequence argument due to Serre and its extension to schemes due to Gabber, we derive that    χ OX OY , OZ = χ OE (OV , OW ).

(2)

If dim Y + dim Z < dim X, then dim V < d and part (a) follows from the above proposition.

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For proof of part (b), first note that since X  = X ⊗K PN K and since TX is generated global by sections, TX is also generated global by sections. Next we appeal to Proposition 2.1. By this, it will be enough to show that (I/I 2 )∨ is generated by global sections. Recall that W = Z  is smooth and hence ΩOW/K is a locally free OW -module. Then we have following exact sequence 0 → I/I 2 → ΩOX /K ⊗OX OW → ΩOW/K → 0

(3)

which is locally split since W is smooth over K. Applying Hom(−, OW ) we obtain our required exact sequence ∨  0 → TW/K → i ∗ TX /K → I/I 2 → 0

(4)

where i : W → X  the natural inclusion. Thus (I/I 2 )∨ is generated by global sections. This completes the proof of our theorem. 2 Remarks. (1) The same proof works if one assumes X is smooth and at least one of Y and Z is projective over K. (2) M. Hochster inquired whether there exists examples showing that part (b) of the above theorem is false when TX is not generated by global sections. We produced such examples on blow-ups of a regular variety at a closed point where supports of both coherent modules are contained in the exceptional divisor. The referee informed us that examples of negative intersection multiplicities on blow-ups of a point in P2 have been known for many years. So we refrain from providing any detail about our examples. 2.4.

Next we prove the following non-negativity on PdR = P.

Theorem 4. Let Y and Z be two closed subschemes of P such that Y ∩ Z ⊂ Pds = PdK . Then (i) if dim Y + dim Z < dim P, χ OP (OY , OZ ) = 0, and (ii) if dim Y + dim Z = dim P, χ OP (OY , OZ )  0. Proof. The proof is obtained by tracing Gabber’s path. Step 1. By using De Jong’s theorem on alteration we construct Z  , X  , where Z  is a regular alter   ation of Z, X  = PdR × PN R , Z is a closed subscheme of X and π : X → P is the projection map. Let Y  = π −1 (Y ). Then, by Grothendieck’s theorem on hypercohomology (EGA III), properness of π and induction on dimension, we have χ OX (OY  , OZ  )  0 according as χ OP (OY , OZ )  0. Step 2. Since Z  is regular, it is a local complete intersection closed subscheme of X  . Let corresponding normal I denote the ideal of definition of Z  in X  and let E  denote the bundle. Due to a spectral sequence argument of Serre, if V  = Spec( t0 I t OY  /I t+1 OY  ), then χ OX (OY  , OZ  ) = χ OE  (OV  , OZ  ). Since mh OV  = 0 for some h > 0, the sign of χ OE  (OV  , OZ  ) is determined by χ OE (F, OW ), where E is the fiber Es of E  over s = [m]

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in Spec R, W = Zs and F represents a quotient in the filtration OV  ⊃ mOV  ⊃ m2 OV  ⊃ · · · ⊃ mh−1 OV  ⊃ (0). Step 3. Let X1 = Spec(R/m2 ), X = Spec R and X1 = X1 ×X X  . Assume R is equicharacteristic or ramified. Can write R/m2 = K[T1 , . . . , Tn ]/(T1 , . . . , Tn )2 (in the mixed characteristic case, the mixed characteristic p ∈ m2 ). We have the following claim: d  → ΩX1/K ⊗ OW splits locally. Claim. I /(I 2 + mI ) −

Proof of the Claim. Let m = (x1 , . . . , xd ) and A = R[ xx21 , . . . , xxn1 ]—the co-ordinate ring of U1 in PdR . Let Z˜ ∩ U1 = Spec(A/q). ˜ Let V1 = Spec A[U1 , . . . , UN ] be an affine open set contained ˜ in OV1 . We have an injection in π −1 (U1 ) and let I denote the ideal of definition of π −1 (Z) A/q˜ → A[U ]/I . By shrinking U1 , V1 , if necessary, we can assume I is generated by a regular sequence which actually forms a regular system of parameters at the stalk of a closed point in the closed fiber Zs over s. Write B = A[U ]/m2 , and C = A[U ]/(I + mA[U ]), then Spec B and Spec C are the corresponding affine open sets in X1 and Zs . Let N denote the maximal ideal in A[U ] corresponding to the closed point mentioned above. The natural map, induced by d, I /I 2 → ΩB/K ⊗ C is such that I /I 2 ⊗ A[U ]/N → (ΩB/K ⊗ C) ⊗ C/N, i.e., I /I 2 ⊗ K → ΩB/K ⊗ K is an injection, since it factors through N/N 2  ΩB/K ⊗ B/N . This completes the proof of our claim.   η1∗ (ΩPd ) ⊕ η2∗ ΩPN , where η1 and η2 are the projections from X1 to PdX1 We have ΩX1/K X1/K

and PN K respectively and ΩPd

X1/K

PdX1

K

= q1∗ (ΩX1/K ) ⊕ q2∗ (ΩPd ), where q1 , q2 are the projections from K

PdK

to X1 and respectively.  =  , OXs ) is generated by global sections. Note that Xs = X1s This shows that HomOX (ΩX1/K

 PdK × PN is a locally free OXs -module and its dual is generated by K is smooth over K; ΩXs/K global sections. We have a locally split short exact sequence:

d → ΩX  0 → mOX − 1

1/K

⊗ OXs → ΩOX

s/K

→ 0.

(1)

It is easy to check, via d and local considerations, that mOX1 is a locally free OXs -module and  ⊗ OXs . so is ΩX1/K  , OXs ) Consider the exact sequence OXs → OW → 0. It follows, from above, that Hom(ΩX1/K → Hom(ΩX , OW ) is surjective. Thus Hom(ΩX , OW ) is generated by global sections. 1/K

1/K

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Hence, by our claim, Hom(I /(I 2 + mI ), OW ) is generated by global sections. Thus E is generated by global sections—and we are done by Proposition 2.1. Step 4. The unramified case follows from the ramified case via the base change R → R[T ]/(p − T 2 ). 2 2.6. Now we would like to determine the effect on intersection multiplicity via pull-back of vector-bundles. Proposition. Let W be a smooth complete scheme over a field K and let ϕ˜ : E  → E be a OW -map between two locally free OW -modules of ranks r  and r respectively. Write E  = Spec(SymOW (E  )) and E = Spec(SymOW (E)) and ϕ : E → E  denote the map of vector bundles corresponding to ϕ. ˜ Let V be a closed subscheme of E  such that the dimension of support OE  of Tori (OV , OE ) < r for i > 0. Let F be a coherent OW -module. Then χ OE  (OV , F) = χ OE (Oϕ −1 (V ) , F). Proof. Since E  is regular, we obtain a resolution ξ• of OV by locally free OE  -modules of finite rank: ξ• : 0 → ξb → ξb−1 → · · · → ξ1 → ξ0 → 0.

(1)

We pull back ξ• via ϕ and obtain a complex of locally free OE -modules of finite rank ϕ ∗ (ξ• ) : 0 → ϕ ∗ (ξb ) → ϕ ∗ (ξb−1 ) → · · · → ϕ ∗ (ξ1 ) → ϕ ∗ (ξ0 ) → 0. OE 

Note that H0 (ϕ ∗ (ξ• )) = ϕ ∗ (OV ) and Hi (ϕ ∗ (ξ• )) = Tori by G•

(2)

(OV , OE ). Let us denote ϕ ∗ (ξ• ) ⊗ F

G• : 0 → ϕ ∗ (ξb ) ⊗OE F → · · · → ϕ ∗ (ξ0 ) ⊗OE F → 0.

(3)

By taking a finite resolution of F over E by locally free OE -modules and then tensoring it with 2 term is TorOE (F, TorOE  (O , O )); this spectral (2) we obtain a spectral sequence whose Ep,q V E p q sequence converges to Hp+q (G• ). Hence 

     O  (−1)i+j dimK H i E, Hj (G• ) = (−1)i χ OE F, Tori E (OV , OE ) . OE 

For j > 0, dim Supp Torj

(OV , OE ) is less than r and hence by Proposition 2.2,

O  χ OE (F, Torj E (OV , OE )) = 0

for j > 0. Thus

     (−1)i+j dimK H i E, Hj (G• ) = χ OE F, ϕ ∗ (OV ) .

(4)

On the other hand, since ϕ : E → E  is affine, R i ϕ∗ (G) = 0 for any quasi-coherent OE module G. Thus if I • is an injective resolution of any quasi-coherent OE -module G, then ϕ∗ (I • ) is a flabby resolution of ϕ∗ (G). Moreover, we know that for any locally free OE  -module L and for any OE -module G, R i ϕ∗ (G) ⊗ L  R i ϕ∗ (G ⊗ ϕ ∗ (L)). In particular H 0 (E  , ϕ∗ (G) ⊗ L) 

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H 0 (E, G ⊗ ϕ ∗ (L)). Also note that ϕ∗ (F) = F . Hence, taking all these things into consideration and tensoring (2) with an injective resolution of F or taking a Cartan–Eilenberg injective resolution of (3), we obtain (Proposition 11.5.3, Chapter 0, EGA III and Lemma (5.5.2) in [Gr1])    (5) (−1)i+j dimK H i E, Hj (G• ) = χ OE  (F, OV ). From (4) and (5) the required result follows.

2

Corollary. Let X be a smooth projective variety over a field K such that TX is generated by global sections. Let W be a smooth closed subvariety of codimension r. Let I denote the ideal of definition of W in X and let E be the corresponding normal cone. Let E  be a vector bundle on W and let V be a closed subvariety of E  . Suppose that there exists a map ϕ : E → E  of O  bundles such that the dimension of support of Tori E (OV , OE ) < r for i > 0. Then, for any coherent sheaf F on W , χ OE  (OV , F) = χ OE (Oϕ −1 (V ) , F). Moreover, if dim ϕ −1 (V )  r, then χ OE  (OV , OW )  0. Proof. The proof follows from the above proposition and the fact that χ OE (Oϕ −1 (V ) , OW )  0— which is a consequence of Proposition 2.1 and a crucial point in the proof of Theorem 4(2.2) which shows that (I/I 2 )∨ is generated by global sections (exact sequence (4) in the proof). Recall that E = Spec(SymOW (I/I 2 )). 2 3. ˜ Notations are as in 1.4. In the proof of our main 3.1. In this section, our main focus is on X. theorem we will use several results from [Fu]. In particular we would like to state the following results (without proof) from Chapters 3 and 12 in [Fu]. Proposition 1. (See Proposition 3.3 in [Fu].) Let E be a vector bundle of rank r on an algebraic scheme X, π : E → X the projection, s = sE denote the 0-section of this bundle. Let β ∈ Ak E, and let β¯ be any element of Ak (P(E ⊕ 1)) which restricts to β in Ak E. Then ¯ where q is the projection from P(E ⊕ 1) to X and ξ is the s ∗ (β)(= (π ∗ )−1 (β)) = q∗ (cr (ξ ) ∩ β) universal rank r quotient bundle of q ∗ (E ⊕ 1). Proposition 2. (See part (a) of Theorem 12.1 in [Fu].) Let E be a vector bundle of rank r on a scheme X, π : E → X the projection, sE : X → E the 0-section. Let V be a k-dimensional  subvariety of E, k  r. If E is generated by sections, then sE∗ [V ] ∈ Ak−r (X). We will also use a deeper aspect of De Jong’s theorem on regular alteration i.e., regular alteration of a pair (Z, Z0 ) by which we mean that the inverse image of Z0 in a regular alteration of Z is a non-reduced strict normal crossing divisor. This plays a crucial role in our proof of part (iii) of Theorem 5. Let us remind the reader of our set-up (described in the introduction). X˜ = blow-up of X (the fiber over s) at s = [m], Y˜ , Z˜ two closed subvarieties of X˜ such that Y˜ ∩ Z˜ ⊂ E = Pn−1 K ˜ Then Z  has a closed immersion in PN = X  ˜ Let Z  be a regular alteration of Z. in X. X˜ for some N > 0. Let π : X  → X˜ be the projection and let Y  = π −1 (Y˜ ). Let I denote the

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defining ideal of Z  in X  and r = codim . Z  in X  . Write E = Spec(SymOZ (I /I 2 )) and  V = Spec( t0 I t OY  /I t+1 OY  ). Let E1 , . . . , Ed ; V1 , . . . , Ve and W1 , . . . , Wd denote the components of Es (fiber over s), V and Zs respectively. Write Eij to denote the restriction of E to Wi ∩ Wj . Theorem 5. Let (R, m, K) denote a regular local ring of dimension n essentially of finite type over a field k or an excellent discrete valuation ring (V , p). Assume k to be perfect. Write X = Spec R. Let X˜ denote the blow-up of X at {m}, E = the exceptional divisor at {m}. Let Y˜ and Z˜ ˜ Then be closed subvarieties of X˜ such that Y˜ ∩ Z˜ ⊂ E and dim Y˜ + dim Z˜  dim X. (i) if dim Y˜ + dim Z˜ < dim X˜ or Y˜ ∩ Z˜ = φ, χ OX˜ (OY˜ , OZ˜ ) = 0. (ii) If Y˜ , Z˜ are the blow-ups of Spec(R/P ), Spec(R/q) respectively with (R/(P + q)) < ∞, dim R/P + dim R/q = dim R and dim G(R/P ) ⊗ G(R/q)  1, then χ OX˜ (OY˜ , OZ˜ )  0. (iii) There exists sub-bundles Fαβ of rank (r − 1) of Eαβ , 1  α, β  d, such that if any Vi is contained in any Fαβ , then χ OE (OVi , OZs )  0. ˜ Proof. Since R is regular local, X˜ is a regular scheme and Y˜ ∩ Z˜ ⊂ E = Pn−1 K . Let EZ˜ = Z ∩ E O O ∗ ˜ E ˜ ˜ ˜ X and let β : E → X. If Y ⊂ E and Z ⊂ E, then χ (OY˜ , OZ˜ ) = χ (β (OY˜ ), OZ˜ ) and the required result follows by Corollary 2, Theorem 1. If both Y˜ and Z˜ are contained in E and dim Y˜ + dim Z˜ < n, then by a spectral sequence argument, we have   χ OX˜ (OY˜ , OZ˜ ) = χ OE (OY˜ , OZ˜ ) − χ OE OY˜ (1), OZ˜ = 0, by Corollary 2, Theorem 1(1.2). Let π  : Z  → Z˜ be a regular alteration [J, Theorem 4.1, Theorem 8-2]. We can assume  Z → Proj(OZ˜ [U1 , . . . , UN +1 ]) and hence Z  → X  = Proj(OX˜ [U1 , . . . , UN +1 ]) is a closed imbedding such that the diagram Z

X

π

Z˜

π

X˜

commutes. Since X  and Z  are both regular schemes, Z  is a local complete intersection in X  of codimension r say. Let I denote the ideal of definition of Z  in X  . Then I is locally defined by a regular sequence of length r. Let Y  = π −1 (Y˜ ). Since X  → X˜ is flat of relative dimension N , dim Y  = dim Y˜ + N . Note that dim Z  = dim Z˜ by construction. Hence dim Y  + dim Z   dim X  ; equality holds if and ˜ Since π is proper, by pulling back a locally free O ˜ -resolution only if dim Y˜ + dim Z˜ = dim X. X of OY˜ via π , we obtain, by projection formula, χ OX (OY  , OZ  ) = χ OX˜ (OY˜ , π∗ OZ  ) +

   (−1)i χ OZ˜ OY˜ , R i π∗ OZ  . i1

(1)

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Since dim support of π∗ OZ  = dim support of OZ˜ (Z  → Z˜ is an alteration) and for i > 0, dim support of R i π∗ OZ  < dim support of OZ˜ , by induction on dimension, it is enough to prove the theorem for χ OX (OY  , OZ  ). Let E = Spec(OX /I ⊕ I/I 2 ⊕ · · ·). Then E is a vector bundle of rank r over Z  . Write W = Z  and let w : W → E be the zero-section. Let V = Spec(OY  /IOY  ⊕ IOY  /I 2 OY  ⊕ · · ·). Then dim E = dim W + r = dim X  and dim V = dim Y  . By a spectral sequence argument due to Serre and its extension to schemes due to Gabber, we have χ OX (OY  , OZ  ) = χ OE (OV , OW ).

(2)

Note that, since Y˜ ∩ Z˜ ⊂ E, ∃t > 0, such that mt OV = 0. Hence by considering a filtration OV ⊃ mOV ⊃ · · · ⊃ mt−1 OV ⊃ 0, we can replace OV by G such that m · G = 0. Let s denote the closed point of X. Let Xs , Es and Ws denote the fibers of X  , E and W over s respectively. Let W1 , . . . , Wd be the irreducible components of Ws . Let Ei be the restriction of Es to Wi . Then E1 , . . . , Ed are the irreducible components of Es . By induction on dimension, we can replace G by an irreducible component Vi of the support of G, contained in Ei for some i, 1  i  d. We have: χ OE (OVi , OW ) = χ OEs (OVi , OWs ) = χ OEi (OWi , OVi ).

(3)

˜ then dim V < r = rank of E; hence dim Vi < r = rank of Ei . If dim Y˜ + dim Z˜ < dim X, Thus, by Proposition 2.2, we have χ OX˜ (OY˜ , OZ˜ ) = 0 in this case. This proof also shows that for any proper closed subscheme Y of X  such that dim Y + dim Z  < dim X  and Y ∩ Z  ⊂ Xs , we have χ OX (OY , OZ  ) = 0. (ii) If dim G(R/P ) ⊗ G(R/q) = 0, then Y˜ ∩ Z˜ = φ. Hence, by (i), χ OX˜ (OY˜ , OZ˜ ) = 0. If ˜ say α1 , . . . , αt . Then dim G(R/P ) ⊗ G(R/q) = 1, then Y˜ ∩ Z˜ is a finite set of closed points in X, t OX,α ˜ O ˜ i (O χ X (OY˜ , OZ˜ ) = i=1 χ ˜ i )  0, by non-negativity part of Serre’s Conjecture Y˜ ,αi , OZ,α due to Gabber. (iii) For part (iii), note that we can construct π  : W = Z  → Z˜ in such a way that Ws =  −1 π (EZ˜ ) is a non-reduced strict normal crossing divisor in Z  , i.e., the reduced part of Ws is a strict normal crossing divisor. Then, for any non-empty subset of α ⊂ {1, . . . , d}, the closed  subscheme Wα = i∈α Wi is a regular subscheme of codimension # α in W . Since K is perfect, Wα is smooth and consequently Ei ’s are also the same. Suppose that V1 ⊂ E12 . Write W12 = W1 ∩ W2 . Then, from (3), χ OE1 (OV1 , OW1 ) = χ OE12 (OV1 , OW12 ).

(4)

∗ E. Let J denote the ideal of deConsider the inclusion α12 : W12 → W . Then E12 = α12 1  ¯ finition of W1 in X ; then J1 = J /mOX is the ideal of definition of W1 in Xs . Note that  be the I + mOX ⊂ J1 . Let E1 denote the normal bundle to the inclusion W1 → Xs and E12 n−1 N   restriction of E1 to W12 . Since Xs = PK × PK , TXs /K is generated by global sections. Since  . W1 is a smooth closed subvariety of Xs , E1 is generated by global sections and hence so is E12  Let J12 denote the ideal of definition of W12 in X . Then the natural map

ψ : I/IJ12 → J1 /(J1 J12 + mOX )

(5)

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 → E , where E = Spec(Sym  leads to a map of bundles ϕ : E12 12 12 OW12 (I /I J12 )) and E12 = Spec(SymOW (J1 /(J1 J12 + mOX ))). 12 We would like to consider ψ , ϕ locally.

Local Picture (in a small affine neighborhood in X  of a closed point in W1 ). Let x be a lo˜ Z). ˜ Let A be a regular domain such that cal equation of E(EZ˜ ) in an affine open set of X(      X = Spec A and Xs = Spec(A /xA ). Write A¯ = A /xA . Let Γ (Spec A , I) = I , where I is generated by an A -sequence of length r; then W = Spec(A /I ). Note that A /xA , A /I are ¯ 1  β  r, is an A-sequence. ¯ Write both regular and if I = (a1 , . . . , ar ) then {a¯ β = Im(aβ ) in A},  ˜ ˜ A = A /I and x˜ = im(x) in A. Since Ws is a strict normal crossing divisor in W , ∃t1 , . . . , td in A such that n

x˜ = t˜1n1 t˜2n2 · · · t˜j j ,

ni  1, 1  i  j, j  d,

(6)

˜ t˜1 , . . . , t˜i ) is a regular ring of codimension i in A. ˜ ˜ where t˜i = im(ti ) in A˜ and A/( in A, nj n2 ˜ ˜ ˜ ˜ ˜ Here W = Spec(A), Ws = Spec(A/x A) and Wi = Spec(A/ti A). Let a = t2 · · · tj . We have ˜ A˜ ⊗ ˜ Sym ˜ (I /I 2 )) = Spec(Sym ˜ ˜ (I /(I 2 + xI ))) E = Spec(Sym ˜ (I /I 2 )), Es = Spec(A/x A

A

A

A/x A

˜ 1 A˜ ⊗ ˜ ˜ Sym ˜ ˜ (I /I 2 + xI )) = Spec(Sym ˜ ˜ (I /(I 2 + t1 I ))). Note and E1 = Spec(A/t A/x A A/x A A/t1 A ˜ t˜i A) ¯ I¯, t¯i )), where I¯ = im(I ) in A, ¯ t¯i = Im(ti ) ˜ = Spec(A /(I, ti )) = Spec(A/( that Wi = Spec(A/ ¯ I¯, t¯1 )) is regular of codimension r + 1(r) in A (A) ¯ and A¯ is regular, ¯ Since A /(I, t1 ) (A/( in A. x is a minimal generator of (I, t1 ) and its image is part of a basis of (I, t1 )/(I, t1 )2 . This implies, by (6), that x − at1n1 = Σχβ aβ , 1  β  r, where one of the λβ is a unit (we may have to shrink Spec A a bit to achieve this). Thus, without any loss of generality, can assume that x − at1n1 = aβ

for some β, 1  β  r. We fix this β.

(7)

Now we consider ψ in (5) locally.   ψ : I / I 2 + (t1 , t2 )I →

(I, t1 ) , (I, t1 )(I, t1 , t2 ) + (x) ψ(Imaj ) = Imaj , j = β,   ψ(Imaβ ) = Imaβ = im x − at1n1 = 0;     ψ˜ : A/ I + (t1 , t2 ) [X1 , . . . Xr ] → A/ I + (t1 , t2 ) [Y1 , . . . Yr ], ˜ i ) = Yi , ψ(X

1  i  r − 1,

˜ r ) = 0, ψ(X Ker ψ˜ = (Xr ).

(8)

Now we would like to prove the following claims. Claim 1. The collection of {Imaβ } in I /(I 2 + (t1 , t2 )I ) defines an effective Cartier divisor D on E12 . Moreover, the support of D is a locally trivial bundle F12 of rank (r − 1) on W12 . The proof of this claim follows easily from (8).

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Claim 2. Let L12 = Ker ψ. Then L12 = π ∗ (OP (1)) where π  : W12 → Pn−1 = P is induced by ˜ the regular alteration map: π  : Z  → Z. Proof of the claim. The proof follows from local considerations. Suppose that m, the maximal ˜ Z) ˜ ideal of R is generated by x1 , . . . , xn . Let D+ (xi ) denote the standard affine open sets of X( and let D+ (x¯i ) denote the standard affine open sets of E = P(EZ˜ ). Let Q be a closed point in π −1 (D+ (x¯i ) ∩ D+ (x¯j )). As in (6), in a small affine neighborhood of Q in W , we have x˜i = t˜1n1 · · · t˜n

and x˜j = s˜1r1 · · · s˜hrh ,

  d, h  d.

(9)

In Γ (D+ (xi xj ), OX˜ ), xj = (xj /xi )xi where xj /xi is a unit. Similar equations hold in Γ (D+ (x¯i x¯j ), OP ). Thus from (9) we have n r t˜1n1 · · · t˜  = (x˜i /x˜j )˜s1r1 · s˜hh .

Since W is regular, we have  = h, nμ = rμ , 1 ⊆ μ  , and there exist units αμ such that tμ − αμ βμ ∈ I for 1  μ  . Since xi = t1n1 · · · tlnl + aβi and xj = s1n1 · · · slnl + aβj (7), we have (xj /xi )aβi − aβj ∈ (I 2 , (t1 , t2 )I ). Hence (x¯j /x¯i ) Im aβi = Im aβj in I /(I 2 , (t1 , t2 )I ). Thus L12 = π  ∗ (OP (1)). Another proof of the above claim, when R is ramified, can be obtained by using module of differentials and the exact sequence (1) in the proof of Theorem 4.  is generated by global sections, it follows from claim 1 that F is also generated Since E12 12 by global sections. Assume that V1 ⊂ F12 . Let i denote the natural inclusions for both F12 → E12 and P(F12 ⊕ 1) → P(E12 ⊕ 1). Denote ∗ (E ∨ ⊕ 1), where q : P(E ⊕ 1) → W by ξ12 the universal quotient bundle of rank r of q12 12 12 12 12 is the projection map. Let q = q12 |P(F12 ⊕1) and ξ = the universal quotient bundle of rank (r − 1) on P(F12 ⊕ 1). This set-up leads to the following exact sequence   0 → ξ → i ∗ (ξ12 ) → q ∗ L∨ 12 → 0.

(10)

We have χ OE12 (OV1 , OW12 ) =



=

=

cr (ξ12 ) ∩ [V¯1 ]

  part (iv), Proposition 2.2

   ¯ c 1 q ∗ L∨ 12 cγ −1 (ξ ) ∩ [V ]   c 1 L∨ 12 ∩ δ,

  exact sequence (10)

  where δ = q∗ cγ −1 (ξ ) ∩ V¯1 .

Since F12 is generated by global sections, by Proposition 2 of this section, δ = s ∗ (V1 ) ∈    c1 (L∨ A1 (W12 ). Since π∗ [c1 (L∨ 12 ) ∩ δ] = c1 (OP (−1)) ∩ π∗ δ, we have 12 ) ∩ δ  0. Thus

χ OE12 (OV , OW12 )  0 and the proof of our theorem is complete.

2

Corollary 1. Let R be a regular local ring of essentially finite type over a field or over a discrete valuation ring whose field of fractions is global and let M, N be two finitely generated R-modules

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such that (M ⊗R N ) < ∞. Suppose that dim M + dim N = dim R and dim G(M) ⊗ G(N ) < 2. Then χ(M, N)  em (M)em (N ). Corollary 2. R as above. Then vanishing part of Serre’s Conjecture is valid for R. Both corollaries follow from Theorem 2 and the above theorem. Remarks. (1) It follows from Claim 2 in the above theorem that there exists a locally free OW12 -module D∨ of rank 1 such that D∨ = π  ∗ (OP (1)) and o → D∨ → I /I J12 → β → 0

(11)

is exact and locally split. This leads to the following locally split exact seqn 0 → β ∨ → (I /I J12 )∨ → D → 0. Since D is not generated by global sections (unless trivial!), (I /I J )∨ cannot be generated by global sections. This implies that HomOW (I /I 2 , OWs ) is not generated by global sections (proof of Theorem 4). Thus Gabber’s technique fails for X˜ even when none of Y˜ , Z˜ is contained in the exceptional divisor (Remark 2 at the end of Theorem 3). (2) In the equicharacteristic case Serre proved that χ(M, N)  em (M)em (N ) [S1, Theorem 1, Chapter V]. Hence, in such a case, in general, all the components V1 , . . . , Ve of V cannot be contained in Fα,β ’s. (3) Part (iii) of Theorem 5 demonstrates the possibility of non-positive intersection multiplicity for certain component of a subcone of a bundle originating from the blow-up situation. However, we have already shown in Corollary 2, Theorem 2, that χ OX˜ (OY˜ , OZ˜ )  0 in the equicharacteristic case, where Y˜ = blow-up of Spec(R/P ) and Z˜ = blow-up of Spec(R/q), by using Serre’s Theorem on intersection multiplicity. Now we would like to present an independent proof of this result. Our proof is inspired by Chapter 12 in [Fu]. Proof. Let Δ : X → X × X (X = Spec R) and Δ˜ : X˜ → X˜ × X˜ be the diagonal maps. Write ˜ Let I denote the kernel of Δ∗ : R ⊗k R → R. The normal bundle for Δ˜ is Ω ˜ M = X˜ × X. X/k and its dual sheaf TX/k is not generated by global sections. By diagonalization argument [S1] ˜ χ OX˜ (OY˜ , OZ˜ ) = χ OM (OY˜ ×Z˜ , OX˜ ).

(1)

Blow-up M at E × E—denote this blow-up by B and let η : B → M be the projection. Then ˜ is η−1 (Y˜ × Z)—denote ˜ the blow-up of Y˜ × Z˜ at (E × E) ∩ (Y˜ × Z) it by N . Since blow-up of ˜ X˜ imbeds into B and this imbedding is regular. Denote it by h. Then X˜ at E = (E × E) ∩ X˜ is X, χ OM (OY˜ ×Z˜ , OX˜ ) = χ OB (ON , OX˜ ).

(2)

Let π : X˜ → X be the projection. It can be shown by local considerations that the normal bundle F for the imbedding h is isomorphic to π ∗ (I /I 2 ) ⊗ O(−E). Hence F is generated by global sections.

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 Let J denote the sheaf of ideals defining X˜ in B. Let C = Spec( t0 J t ON /J t+1 ON ). Then χ OB (ON , OX˜ ) = χ OF (OC , OX˜ ). The right-hand side of (3) is non-negative by Proposition 2.1.

(3)

2

Acknowledgments I am thankful to O. Gabber and W. Fulton for pointing out misconceptions that I had in the development of this material. I would also like to acknowledge Gabber for going through Sections 1, 2 and part of Section 3 of this paper. References [B] [D1] [D5] [D-H-M] [Fu-M] [Fu] [G] [G-S] [Gr1] [H] [Ho2] [J] [Ma] [P-S2] [R1] [S1] [S2]

P. Berthelot, Altérnations de variétés algébriques [d’après A.J. de Jong], Seminaire Bourbaki 815 (1996). S.P. Dutta, Generalized intersection multiplicity of modules, Trans. Amer. Math. Soc. 276 (1983) 657–669. S.P. Dutta, A special case of positivity II, Proc. Amer. Math. Soc. 133 (7) (2005) 1891–1896. S.P. Dutta, M. Hochster, J.E. McLaughlin, Modules of finite projective dimension with negative intersection multiplicities, Invent. Math. 79 (1985) 253–291. W. Fulton, R. MacPherson, Intersecting cycles on an algebraic variety, in: P. Holm (Ed.), Real and Complex Singularities, Oslo, Sijthoff and Noordhoff International Publishers, 1976. W. Fulton, Intersection Theory, Springer-Verlag, Berlin, 1984. O. Gabber, Non-negativity of Serre’s intersection multiplicities, exposé à L’IHES, décembre 1995. H. Gillet, S. Soulé, K théorie et nullité des multiplicités d’intersection, C. R. Acad. Sci. Paris Sér. I 300 (1985) 71–74. A. Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. IX (1957) 119–221. R. Hartshorne, Algebraic Geometry, 6th corrected printing, Springer-Verlag, 1993. M. Hochster, Nonnegativity of intersection multiplicities in ramified regular local rings following Gabber/ de Jong/Berthelot, preprint. A.J. de Jong, Smoothness, semi-stability and alterations, Publ. Math. Inst. Hautes Études Sci. 83 (1996) 51–93. H. Matsumura, Commutative Algebra, second edition, Benjamin/Cummings Publishing Company, Reading, MA, 1980. C. Peskine, L. Szpiro, Sygyzies et multiplicités, C. R. Acad. Sci. Paris 278 (1974) 1421–1424. P. Roberts, The vanishing of intersection multiplicities of perfect complexes, Bull. Amer. Math. Soc. 13 (1985) 127–130. J.-P. Serre, Algèbre Locale: Multiplicités, 3rd ed., Lecture Notes in Math., vol. 11, Springer-Verlag, Berlin/Heidelberg/New York, 1975. J.-P. Serre, Faisceaux algébriques cohérents, Ann. of Math. LXI (1955) 197–278.

Further reading [D-Gr] J. Dieudonné, A. Grothendieck, Eléments de géométrie algébrique (EGA) I, Publ. Math. Inst. Hautes Études Sci. 4 (1960); II, Publ. Math. Inst. Hautes Études Sci. 8 (1961); III, Publ. Math. Inst. Hautes Études Sci. 11 (1961); Publ. Math. Inst. Hautes Études Sci. 17 (1963); IV, Publ. Math. Inst. Hautes Études Sci. 20 (1964); Publ. Math. Inst. Hautes Études Sci. 24 (1965); Publ. Math. Inst. Hautes Études Sci. 28 (1966); Publ. Math. Inst. Hautes Études Sci. 32 (1967). [D2] S.P. Dutta, Frobenius and multiplicities, J. Algebra 85 (2) (1983) 424–448. [D3] S.P. Dutta, A special case of positivity, Proc. Amer. Math. Soc. 103 (2) (1988). [D4] S.P. Dutta, A theorem on smoothness—Bass–Quillen, Chow groups and intersection multiplicity of Serre, Trans. Amer. Math. Soc. 352 (2) (1999) 1635–1645. [D6] S.P. Dutta, On negativity of higher Euler characteristics, Amer. J. Math. 126 (2004) 1341–1354. [Fo] H.-B. Foxby, The MacRae invariant, in: Commutative Algebra, Durham, 1981, in: London Math. Soc. Lecture Note Ser., vol. 72, 1982, pp. 121–128.

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[Ho1] M. Hochster, Topics in the Homological Theory of Modules over Commutative Rings, CBMS Reg. Conf. Ser. Math., vol. 24, Amer. Math. Soc., 1975. [K] E. Kunz, Characterization of regular local rings of characteristics p, Amer. J. Math. 41 (1969) 772–784. [K-R] K. Kurano, P. Roberts, The positivity of intersection multiplicities and symbolic power of prime ideals, Compos. Math. 122 (2) (2000) 165–182. [L] M. Levine, Localization on singular varieties, Invent. Math. 91 (1988) 423–464. [Li] S. Lichtenbaum, On the vanishing of tor in regular local rings, Illinois J. Math. 10 (1966) 220–226. [M] M.P. Malliavain-Bramaret, Une remarque sur les anneaux locaux réguliers, Sém. Dubriel-Pisot (Algèbre et Théorie des Nombres) 13 (1970/1971). [M-S] C. Miller, A. Singh, Intersection multiplicities over Gorenstein rings, Math. Ann. 317 (2000) 155–171. [P-S1] C. Peskine, L. Szpiro, Dimension projective finie et cohomologie locale, Publ. Math. Inst. Hautes Études Sci. 42 (1973) 49–119. [R2] P. Roberts, Multiplicities and Chern Classes in Local Algebra, Cambridge University Press, 1998. [R-S] P. Roberts, V. Srinivas, Modules of finite length and finite projective dimension, Invent. Math. 151 (2003) 1–28. [Sa] S. Sather-Wagstaff, Intersection of symbolic power of prime ideals, J. London Math. Soc. (2) 65 (3) (2002) 560–574. [Se] G. Seibert, Complexes with homology of finite length and Frobenius functors, J. Algebra 125 (1989) 278–287. [T] B.R. Tennison, Intersection multiplicities and tangent cones, Math. Proc. Cambridge Philos. Soc. 85 (1979) 33– 42.